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\title {DARBOUX-LIKE FUNCTIONS ON ${\mathbb R}^n$ WITHIN THE CLASSES
OF BAIRE ONE, BAIRE TWO, AND ADDITIVE FUNCTIONS}
\author{Krzysztof Ciesielski, Department of Mathematics, West Virginia
University,
Morgantown, WV 26506-6310, email: \tt{KCies@wvnvms.wvnet.edu}}
\markboth{Santa Barbara Symposium -- K.~Ciesielski}{Santa Barbara Symposium
-- K.~Ciesielski}
\begin{document}\label{v-conf-kc}
\maketitle
The talk was based mainly on my joint paper with Jan Jastrz{\c{e}}bski.
In the paper we present an exhaust discussion of
the relations between Darboux-like functions within the classes of Baire one,
Baire two, and additive functions from $\real^n$ into $\real$.
In particular we
construct an additive extendable discontinuous function
$f\colon\real\to\real$, answering a question of Gibson and Natkaniec,
and show that there is no similar function from $\real^2$ into $\real$.
We also describe a Baire class two
almost continuous function $f\colon\real\to\real$ which is not extendable.
This gives a negative answer to a problem of Brown, Humke, and Laczkovich.
Some part of the talk was also based on my joint paper
{\em Two examples concerning almost continuous functions} written with
Andrzej Ros{\l}anowski.
In this paper we construct, under the assumption that
union of less than continuum many meager subsets of $\real$ is meager in
$\real$,
an additive connectivity function $f\colon\real\to\real$
with Cantor intermediate value property which is not almost
continuous. This gives a partial answer to a question of
D.~Banaszewski.
We also show that every extendable function $g\colon\real\to\real$
with a dense graph satisfies the following stronger version of the SCIVP
property: for every $a